cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A140156 a(1)=1, a(n) = a(n-1) + n^3 if n odd, a(n) = a(n-1) + n^5 if n is even.

Original entry on oeis.org

1, 33, 60, 1084, 1209, 8985, 9328, 42096, 42825, 142825, 144156, 392988, 395185, 933009, 936384, 1984960, 1989873, 3879441, 3886300, 7086300, 7095561, 12249193, 12261360, 20223984, 20239609, 32120985, 32140668, 49351036, 49375425
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Magma
    [(1/48)*(9*(-1 +(-1)^n) + 4*(1 -12(-1)^n)*n^2 + 12*(1 -(-1)^n)*n^3 + (16 + 30*(-1)^n)*n^4 + 12*(1 +(-1)^n)*n^5 + 4*n^6): n in [1..50]]; // G. C. Greubel, Jul 05 2018
  • Mathematica
    a = {}; r = 3; s = 5; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (* Artur Jasinski *)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^3,a+(n+1)^5]}; Transpose[ NestList[ nxt,{1,1},30]][[2]] (* or *) LinearRecurrence[ {1,6,-6,-15, 15,20,-20,-15,15,6,-6,-1,1},{1,33,60,1084,1209,8985,9328, 42096, 42825, 142825,144156, 392988,395185},40] (* Harvey P. Dale, Aug 27 2013 *)
Table[(1/48)*(9*(-1 +(-1)^n) + 4*(1 -12(-1)^n)*n^2 + 12*(1 -(-1)^n)*n^3 + (16 + 30*(-1)^n)*n^4 + 12*(1 +(-1)^n)*n^5 + 4*n^6), {n, 1, 50}] (* G. C. Greubel, Jul 05 2018 *)
  • PARI
    for(n=1, 50, print1((1/48)*(9*(-1 +(-1)^n) + 4*(1 -12(-1)^n)*n^2 + 12*(1 -(-1)^n)*n^3 + (16 + 30*(-1)^n)*n^4 + 12*(1 +(-1)^n)*n^5 + 4*n^6), ", ")) \\ G. C. Greubel, Jul 05 2018

Formula

G.f.: -x*(1 + 32*x + 21*x^2 + 832*x^3 - 22*x^4 + 2112*x^5 - 22*x^6 + 832*x^7 + 21*x^8 + 32*x^9 + x^10)/((1+x)^6*(x-1)^7). - R. J. Mathar, Feb 22 2009

A140157 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^0 if n is even.

Original entry on oeis.org

1, 2, 83, 84, 709, 710, 3111, 3112, 9673, 9674, 24315, 24316, 52877, 52878, 103503, 103504, 187025, 187026, 317347, 317348, 511829, 511830, 791671, 791672, 1182297, 1182298, 1713739, 1713740, 2421021, 2421022, 3344543, 3344544, 4530465
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Magma
    [(1/60)*(15*(-1 + (-1)^n) + (29 +15*(-1)^n)*n + 10*(1 -3*(-1)^n)*n^3 + 15*(1 -(-1)^n)*n^4 + 6*n^5): n in [1..50]]; // G. C. Greubel, Jul 05 2018
  • Mathematica
    a = {}; r = 4; s = 0; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (* Artur Jasinski *)
LinearRecurrence[{1,5,-5,-10,10,10,-10,-5,5,1,-1}, {1, 2, 83, 84, 709, 710, 3111, 3112, 9673, 9674, 24315}, 50] (* or *) Table[(1/60)*(15*(-1 + (-1)^n) + (29 +15*(-1)^n)*n + 10*(1 -3*(-1)^n)*n^3 + 15*(1 -(-1)^n)*n^4 + 6*n^5), {n,1,50}] (* G. C. Greubel, Jul 05 2018 *)
  • PARI
    for(n=1,50, print1((1/60)*(15*(-1 + (-1)^n) + (29 +15*(-1)^n)*n + 10*(1 -3*(-1)^n)*n^3 + 15*(1 -(-1)^n)*n^4 + 6*n^5), ", ")) \\ G. C. Greubel, Jul 05 2018

Formula

G.f.: x*(1 + x + 76*x^2 - 4*x^3 + 230*x^4 + 6*x^5 + 76*x^6 - 4*x^7 + x^8 + x^9)/((1+x)^5*(x-1)^6). - R. J. Mathar, Feb 22 2009

A140158 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^1 if n is even.

Original entry on oeis.org

1, 3, 84, 88, 713, 719, 3120, 3128, 9689, 9699, 24340, 24352, 52913, 52927, 103552, 103568, 187089, 187107, 317428, 317448, 511929, 511951, 791792, 791816, 1182441, 1182467, 1713908, 1713936, 2421217, 2421247, 3344768, 3344800, 4530721
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Magma
    [(1/120)*(15*(-1 +(-1)^n) + (28 + 60*(-1)^n)*n + 30*n^2 + 20*(1 - 3*(-1)^n)*n^3 + 30*(1 -(-1)^n)*n^4 + 12*n^5): n in [1..50]]; // G. C. Greubel, Jul 05 2018
  • Mathematica
    a = {}; r = 4; s = 1; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (* Artur Jasinski *)
LinearRecurrence[{1,5,-5,-10,10,10,-10,-5,5,1,-1}, {1, 3, 84, 88, 713, 719, 3120, 3128, 9689, 9699, 24340}, 50] (* or *) Table[(1/120)*(15*(-1 +(-1)^n) + (28 + 60*(-1)^n)*n + 30*n^2 + 20*(1 - 3*(-1)^n)*n^3 + 30*(1 -(-1)^n)*n^4 + 12*n^5), {n,1,50}] (* G. C. Greubel, Jul 05 2018 *)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^4,a+n+1]}; NestList[nxt,{1,1},40][[;;,2]] (* Harvey P. Dale, Dec 28 2024 *)
  • PARI
    for(n=1,50, print1((1/120)*(15*(-1 +(-1)^n) + (28 + 60*(-1)^n)*n + 30*n^2 + 20*(1 - 3*(-1)^n)*n^3 + 30*(1 -(-1)^n)*n^4 + 12*n^5), ", ")) \\ G. C. Greubel, Jul 05 2018

Formula

G.f.: x*(1 + 2*x + 76*x^2 - 6*x^3 + 230*x^4 + 6*x^5 + 76*x^6 - 2*x^7 + x^8)/((1+x)^5*(x-1)^6). - R. J. Mathar, Feb 22 2009

A140159 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^2 if n is even.

Original entry on oeis.org

1, 5, 86, 102, 727, 763, 3164, 3228, 9789, 9889, 24530, 24674, 53235, 53431, 104056, 104312, 187833, 188157, 318478, 318878, 513359, 513843, 793684, 794260, 1184885, 1185561, 1717002, 1717786, 2425067, 2425967, 3349488, 3350512, 4536433
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Magma
    [(1/60)*(n +n^2)* (4 + 30*(-1)^n + (11 -15*(-1)^n)*n + (9 -15*(-1)^n)*n^2 + 6*n^3): n in [1..50]]; // G. C. Greubel, Jul 05 2018
  • Mathematica
    a = {}; r = 4; s = 2; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^4,a+(n+1)^2]}; NestList[nxt, {1, 1}, 40][[All, 2]] (* Harvey P. Dale, Sep 21 2016 *)
LinearRecurrence[{1,5,-5,-10,10,10,-10,-5,5,1,-1}, {1, 5, 86, 102, 727, 763, 3164, 3228, 9789, 9889, 24530}, 50] (* or *) Table[(1/60)*(n +n^2)* (4 + 30*(-1)^n + (11 -15*(-1)^n)*n + (9 -15*(-1)^n)*n^2 + 6*n^3), {n,1, 50}] (* G. C. Greubel, Jul 05 2018 *)
  • PARI
    for(n=1,50, print1((1/60)*(n +n^2)* (4 + 30*(-1)^n + (11 -15*(-1)^n)*n + (9 -15*(-1)^n)*n^2 + 6*n^3), ", ")) \\ G. C. Greubel, Jul 05 2018

Formula

G.f.: x*(1+4*x+76*x^2-4*x^3+230*x^4-4*x^5+76*x^6+4*x^7+x^8)/((1+x)^5*(x-1)^6). - R. J. Mathar, Feb 22 2009

A140160 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^3 if n is even.

Original entry on oeis.org

1, 9, 90, 154, 779, 995, 3396, 3908, 10469, 11469, 26110, 27838, 56399, 59143, 109768, 113864, 197385, 203217, 333538, 341538, 536019, 546667, 826508, 840332, 1230957, 1248533, 1779974, 1801926, 2509207, 2536207, 3459728, 3492496
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Magma
    [(1/240)*(15*(1 -(-1)^n) - 4*(1 - 15*(-1)^n)*n + 30*(1 + 3(-1)^n)*n^2 + 20*(5 - 3*(-1)^n)*n^3 + 30*(3 - 2*(-1)^n)*n^4 + 24*n^5): n in [1..50]]; // G. C. Greubel, Jul 05 2018
  • Mathematica
    a = {}; r = 4; s = 3; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (* Artur Jasinski *)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^4,a+(n+1)^3]}; NestList[nxt,{1,1},40][[All,2]] (* or *) LinearRecurrence[{1,5,-5,-10,10,10,-10,-5,5,1,-1},{1,9,90,154,779,995,3396,3908,10469,11469,26110},40] (* Harvey P. Dale, Oct 05 2016 *)
Table[(1/240)*(15*(1 -(-1)^n) - 4*(1 - 15*(-1)^n)*n + 30*(1 + 3(-1)^n)*n^2 + 20*(5 - 3*(-1)^n)*n^3 + 30*(3 - 2*(-1)^n)*n^4 + 24*n^5), {n,1,50}] (* G. C. Greubel, Jul 05 2018 *)
  • PARI
    for(n=1,50, print1((1/240)*(15*(1 -(-1)^n) - 4*(1 - 15*(-1)^n)*n + 30*(1 + 3(-1)^n)*n^2 + 20*(5 - 3*(-1)^n)*n^3 + 30*(3 - 2*(-1)^n)*n^4 + 24*n^5), ", ")) \\ G. C. Greubel, Jul 05 2018

Formula

G.f.: x*(1 + 8*x + 76*x^2 + 24*x^3 + 230*x^4 - 24*x^5 + 76*x^6 - 8*x^7 + x^8)/((1+x)^5*(x-1)^6). - R. J. Mathar, Feb 22 2009

A140161 a(1)=1, a(n) = a(n-1) + n^4 if n odd, a(n) = a(n-1) + n^5 if n is even.

Original entry on oeis.org

1, 33, 114, 1138, 1763, 9539, 11940, 44708, 51269, 151269, 165910, 414742, 443303, 981127, 1031752, 2080328, 2163849, 4053417, 4183738, 7383738, 7578219, 12731851, 13011692, 20974316, 21364941, 33246317, 33777758, 50988126
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Magma
    [(1/120)*(15*(-1 +(-1)^n) - 2*(1 -15*(-1)^n)*n - 5*(1 +15*(-1)^n)*n^2 + 20*(1 -3*(-1)^n)*n^3 + (55 + 45*(-1)^n)*n^4 + (42 +30*(-1)^n)*n^5 + 10*n^6): n in [1..50]]; // G. C. Greubel, Jul 05 2018
  • Mathematica
    a = {}; r = 4; s = 5; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (* Artur Jasinski *)
next[{a_,b_}]:={a+1,If[OddQ[a+1],b+(a+1)^4,b+(a+1)^5]}; Transpose[ NestList[ next[#]&,{1,1},30]][[2]] (* Harvey P. Dale, Nov 23 2011 *)
Table[(1/120)*(15*(-1 +(-1)^n) - 2*(1 -15*(-1)^n)*n - 5*(1 +15*(-1)^n)*n^2 + 20*(1 -3*(-1)^n)*n^3 + (55 + 45*(-1)^n)*n^4 + (42 +30*(-1)^n)*n^5 + 10*n^6), {n,1,50}] (* G. C. Greubel, Jul 05 2018 *)
  • PARI
    for(n=1, 50, print1((1/120)*(15*(-1 +(-1)^n) - 2*(1 -15*(-1)^n)*n - 5*(1 +15*(-1)^n)*n^2 + 20*(1 -3*(-1)^n)*n^3 + (55 + 45*(-1)^n)*n^4 + (42 +30*(-1)^n)*n^5 + 10*n^6), ", ")) \\ G. C. Greubel, Jul 05 2018

Formula

G.f.: x*(-1 - 32*x - 75*x^2 - 832*x^3 - 154*x^4 - 2112*x^5 + 154*x^6 - 832*x^7 + 75*x^8 - 32*x^9 + x^10)/((1+x)^6*(x-1)^7). - R. J. Mathar, Feb 22 2009

A140162 a(1)=1, a(n) = a(n-1) + n^5 if n odd, a(n) = a(n-1) + n^0 if n is even.

Original entry on oeis.org

1, 2, 245, 246, 3371, 3372, 20179, 20180, 79229, 79230, 240281, 240282, 611575, 611576, 1370951, 1370952, 2790809, 2790810, 5266909, 5266910, 9351011, 9351012, 15787355, 15787356, 25552981, 25552982, 39901889, 39901890, 60413039
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Magma
    [(1/24)*(3*(-1 +(-1)^n) + 12*n + (-1 +15*(-1)^n)*n^2 + 5*(1 -3* (-1)^n)*n^4 - 6*(-1 +(-1)^n)*n^5 + 2*n^6): n in [1..50]]; // G. C. Greubel, Jul 05 2018
  • Mathematica
    a = {}; r = 5; s = 0; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (* Artur Jasinski *)
LinearRecurrence[{1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1},{1,2,245,246, 3371,3372,20179,20180,79229,79230,240281,240282,611575},40]  (* Harvey P. Dale, Apr 21 2011 *)
  • PARI
    for(n=1,50, print1((1/24)*(3*(-1 +(-1)^n) + 12*n + (-1 +15*(-1)^n)*n^2 + 5*(1 -3* (-1)^n)*n^4 - 6*(-1 +(-1)^n)*n^5 + 2*n^6), ", ")) \\ G. C. Greubel, Jul 05 2018

Formula

G.f.: x*(-1 - x - 237*x^2 + 5*x^3 - 1682*x^4 - 10*x^5 - 1682*x^6 + 10*x^7 - 237*x^8 - 5*x^9 - x^10 + x^11)/((1+x)^6*(x-1)^7). - R. J. Mathar, Feb 22 2009

A140163 a(1)=1, a(n) = a(n-1) + n^5 if n odd, a(n) = a(n-1) + n if n is even.

Original entry on oeis.org

1, 3, 246, 250, 3375, 3381, 20188, 20196, 79245, 79255, 240306, 240318, 611611, 611625, 1371000, 1371016, 2790873, 2790891, 5266990, 5267010, 9351111, 9351133, 15787476, 15787500, 25553125, 25553151, 39902058, 39902086, 60413235
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Magma
    [(1/24)*(n +n^2)*(6*(1 +(-1)^n) - (1-9*(-1)^n)*n + (1 -9*(-1)^n)*n^2 + (4 -6*(-1)^n)*n^3 + 2*n^4): n in [1..50]]; // G. C. Greubel, Jul 05 2018
  • Maple
    a:=proc(n) option remember: if n=1 then 1 elif modp(n,2)<>0 then procname(n-1)+n^5 else procname(n-1)+n; fi: end; seq(a(n),n=1..30); # Muniru A Asiru, Jul 07 2018
  • Mathematica
    Table[(1/24)*(n +n^2)*(6*(1 +(-1)^n) - (1-9*(-1)^n)*n + (1 -9*(-1)^n)*n^2 + (4 -6*(-1)^n)*n^3 + 2*n^4), {n, 1, 50}] (* or *) LinearRecurrence[{1, 6, -6, -15, 15, 20, -20, -15, 15, 6, -6, -1, 1}, {1, 3, 246, 250, 3375, 3381, 20188, 20196, 79245, 79255, 240306, 240318, 611611}, 60] (* G. C. Greubel, Jul 05 2018 *)
  • PARI
    for(n=1,50, print1((1/24)*(n +n^2)*(6*(1 +(-1)^n) - (1-9*(-1)^n)*n + (1 -9*(-1)^n)*n^2 + (4 -6*(-1)^n)*n^3 + 2*n^4), ", ")) \\ G. C. Greubel, Jul 05 2018

Formula

G.f.: -x*(1 + 2*x + 237*x^2 - 8*x^3 + 1682*x^4 + 12*x^5 + 1682*x^6 - 8*x^7 + 237*x^8 + 2*x^9 + x^10)/((1+x)^6*(x-1)^7). - R. J. Mathar, Feb 22 2009
a(n) = (1/24)*(n + n^2)*(6*(1 + (-1)^n) - (1 - 9*(-1)^n)*n + (1 - 9*(-1)^n)*n^2 + (4 - 6*(-1)^n)*n^3 + 2*n^4). - G. C. Greubel, Jul 05 2018

A140143 a(1)=1, a(n)=a(n-1)+n^0 if n odd, a(n)=a(n-1)+ n^5 if n is even.

Original entry on oeis.org

1, 33, 34, 1058, 1059, 8835, 8836, 41604, 41605, 141605, 141606, 390438, 390439, 928263, 928264, 1976840, 1976841, 3866409, 3866410, 7066410, 7066411, 12220043, 12220044, 20182668, 20182669, 32064045, 32064046
Offset: 1

Views

Author

Artur Jasinski, May 12 2008, corrected May 17 2008

Keywords

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Mathematica
    a = {}; r = 0; s = 5; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)

Formula

a(n)=a(n-1)+6a(n-2)-6a(n-3)-15a(n-4)+15a(n-5)+20a(n-6)-20a(n-7)-15a(n-8)+15a(n-9)+6a(n-10)-6a(n-11)-a(n-12)+a(n-13). G.f.: x*(-1-32*x+5*x^2-832*x^3-10*x^4-2112*x^5+10*x^6-832*x^7-5*x^8-32*x^9+x^10 )/((1+x)^6*(x-1)^7). [From R. J. Mathar, Feb 22 2009]

A140150 a(1)=1, a(n)=a(n-1)+n^2 if n odd, a(n)=a(n-1)+ n^4 if n is even.

Original entry on oeis.org

1, 17, 26, 282, 307, 1603, 1652, 5748, 5829, 15829, 15950, 36686, 36855, 75271, 75496, 141032, 141321, 246297, 246658, 406658, 407099, 641355, 641884, 973660, 974285, 1431261, 1431990, 2046646, 2047487, 2857487, 2858448, 3907024, 3908113
Offset: 1

Views

Author

Artur Jasinski, May 12 2008

Keywords

Links

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.

Programs

  • Mathematica
    a = {}; r = 2; s = 4; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a (*Artur Jasinski*)
nxt[{n_,a_}]:={n+1,If[EvenQ[n],a+(n+1)^2,a+(n+1)^4]}; NestList[nxt,{1,1},40][[All,2]] (* or *) LinearRecurrence[{1,5,-5,-10,10,10,-10,-5,5,1,-1},{1,17,26,282,307,1603,1652,5748,5829,15829,15950},40] (* Harvey P. Dale, Aug 28 2017 *)

Formula

G.f.: x*(1+16*x+4*x^2+176*x^3-10*x^4+176*x^5+4*x^6+16*x^7+x^8)/((1+x)^5*(x-1)^6). [From R. J. Mathar, Feb 22 2009]
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